294 XIV INTEGRATION IN LOCALLY COMPACT GROUPS
(ii) Ifp = 1 or 2 and ge JS?P(G), the sequence of norms Np((/n * g) - g)
tends to 0 as n tends to + 00. For each function g e Lg3(G), the sequence
((fn *#)~) converges weakly to g in L£(G), considered as the dual of L£(G)
(12.15 and 13.17).
(iii) Suppose in addition that the supports of the fn are contained in a
fixed compact subset of G, Then for each measure \JL on G, the sequence of
measures
converges vaguely to /* (13.4).
(i) For each x e G and each compact neighborhood V of e we have
from the definitions
- f /.(
f/«
-i
Cv
Let V0 be a compact neighborhood of e and let L be a compact subset of
G. Since V^"1L is compact (12.10.5), the restriction of g to VQ JL is uniformly
continuous with respect to a right-invariant distance on G (3.16.5). Hence,
for each e > 0, there exists a compact neighborhood V c V0 of e such that
\ff(x)-~9($~ix)\^£ for all xeL and all seV. Now choose n0 so that
JCy \ftt(s)\ dp(s) ^ e and |1 - jfn(s) dp(s)\ g e for all n ^ n0. We have then
1-
and therefore, for all x e L,
e,
Cv
where ^ = sup ) \ftt($)\ dft(s). Hence
' with respect